Properties

Degree 1
Conductor 311
Sign $0.0565 - 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.688 + 0.724i)2-s + (0.688 − 0.724i)3-s + (−0.0506 + 0.998i)4-s + (0.979 − 0.201i)5-s + 6-s + (−0.954 − 0.299i)7-s + (−0.758 + 0.651i)8-s + (−0.0506 − 0.998i)9-s + (0.820 + 0.571i)10-s + (−0.918 − 0.394i)11-s + (0.688 + 0.724i)12-s + (−0.0506 − 0.998i)13-s + (−0.440 − 0.897i)14-s + (0.528 − 0.848i)15-s + (−0.994 − 0.101i)16-s + (−0.820 − 0.571i)17-s + ⋯
L(s,χ)  = 1  + (0.688 + 0.724i)2-s + (0.688 − 0.724i)3-s + (−0.0506 + 0.998i)4-s + (0.979 − 0.201i)5-s + 6-s + (−0.954 − 0.299i)7-s + (−0.758 + 0.651i)8-s + (−0.0506 − 0.998i)9-s + (0.820 + 0.571i)10-s + (−0.918 − 0.394i)11-s + (0.688 + 0.724i)12-s + (−0.0506 − 0.998i)13-s + (−0.440 − 0.897i)14-s + (0.528 − 0.848i)15-s + (−0.994 − 0.101i)16-s + (−0.820 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0565 - 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0565 - 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.0565 - 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (165, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ 0.0565 - 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.742046027 - 1.646093137i$
$L(\frac12,\chi)$  $\approx$  $1.742046027 - 1.646093137i$
$L(\chi,1)$  $\approx$  1.618269421 - 0.1342370709i
$L(1,\chi)$  $\approx$  1.618269421 - 0.1342370709i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−25.55968112700289672600981722261, −24.274687866973200706103686710359, −23.20476939990393420543692475772, −22.16099523940121868639980445709, −21.6008995430054450164756538094, −21.01543269885080404082707719866, −20.0191104622566623204496787005, −19.15830397654806744552756245977, −18.40911015372489161919837836478, −16.962221683820162329120184185204, −15.80366866575045991867156431656, −15.02704431242369889531943452723, −14.16162917308624259811986430153, −13.196961095466930293760737073281, −12.743880301710076496302070487960, −11.05541431273809981275024392017, −10.34836985893749565727045098954, −9.483607389169719379053469567157, −8.86779888742186395016469732364, −6.921994975670439118935218384333, −5.855007488480452695771818227875, −4.82334949175037968257293536861, −3.73014549080963474174333725600, −2.58981742717746583055796997469, −1.96243565810854238920017463820, 0.44843736718068453478041273626, 2.493704908930935231058952097087, 3.03562809027860185334098058077, 4.59942379861015539919293699007, 5.92389539578263749707110965303, 6.54105695050760688167022215084, 7.60923497851743271763428751368, 8.61483785769817087095180149403, 9.515464747843062295575264021768, 10.869349825421813257728536120711, 12.69627254378958700875274567069, 12.96015177499090470442834462943, 13.608411348566470317852668629519, 14.59933302736584569236632740885, 15.569761431783170687414038749742, 16.52568233638524356351969505019, 17.608696780846954659295963062279, 18.22361307298582897255762241134, 19.47699880383206155646873197887, 20.5374003592825538586282819254, 21.196212184256308521265320554683, 22.307320773987680308402264032458, 23.157081867415976125193182975785, 24.02384768029997456525858394251, 25.0065777590879546854855079667

Graph of the $Z$-function along the critical line